smoke to use gradient from red to violet through rainbow
Here's a variant of the 'Quick Smoke' material (as created by selecting the default cube, pressing Space and typing 'Quick Smoke' Enter).



The differences are the addition of the Emission shader to make the smoke glow (this would normally be the 'flames') and the additional of a Color Ramp set to vary based on the Generated 'Z' coordinate (adjust the color ramp colors to get the desired 'rainbow').

Note that setting the smoke domain objects to be 'Smoke Adaptive Domain' the smoke rainbow will dynamically span the height of the extent of the smoke.